Linear wave estimates

[Thanks to Chengbo Wang for some corrections \u2013 Ed.]

• Fixed-time estimates for free solutions f :
  • (Energy estimate) If $f(0)$ is in $H^{s}$, then $f(t)$ is also.
  • (Decay estimate) If $f(0)$ has more than $(d+1)/2$ derivatives in $L^{1}$, then $\|f(t)\|_{L^{\infty }}$ decays like $<t>^{-(d-1)/2}$. One can obtain the endpoint of $(d+1)/2$ derivatives if one is willing to localize in frequency or use Hardy spaces and BMO.
  • One can interpolate between these estimates to get $(L^{p},L^{p'})$ estimates with the sharp loss of regularity Br1975. This is useful for Strichartz estimates and for scattering theory.
• Strichartz estimates: A free ${\dot {H}}^{s}$ solution is in $L_{t}^{q}L_{x}^{r}$ if
  • (Scaling) $d/2-s=1/q+d/r$
  • (Parallel interactions) $(d-1)/4\geq 1/q+(d-1)/2r$
  • (Increase of integrability) $q,r\geq 2$
  • (No double endpoints) $(n,q,r)\neq (3,2,\infty )$
    • This estimate can be recovered for radial functions KlMa1993, or when a small amount of smoothing (either in the Sobolev sense, or in relaxing the integrability) in the radial variable [MacNkrNaOz-p].However in the general case one cannot recover the estimate even if one uses the BMO norm or attempts Littlewood-Paley frequency localization Mo1998
    • Actually even when $n>3$, the $(q,r)=(2,infty)$ estimate is slightly subtle; one has BMO and Besov space estimates but not directly $L^{\infty }$ estimates. However, the endpoint $(q,r)=(2,2(d-1)/(d-3))$ is OK; see KeTa1998.
  • In the case $s=1/2,d=3,q=r=4$, a maximizer exists (e.g. with initial position zero and initial velocity given by the Cauchy distribution $1/1+|x|^{2})$, witih best constant $(3pi/4)^{1/4}$ [Fc-p4]
  • These results extend globally outside of a convex obstacle [Bu-p], SmhSo1995, [SmhSo-p], [Met-p]; see [So-p] for a survey of this issue and applications to nonlinear wave equations outside of an obstacle.
  • For data which is radial (or otherwise enjoys additional angular regularity) a much larger range of Strichartz estimates is possible (basically because the parallel interaction obstruction is substantially weakened); see [Stz-p4] for further discussion.

These estimates extend to some extent to the Klein-Gordon equation $Boxu=m^{2}u$.A useful heuristic to keep in mind is that this equation behaves like the Schrodinger equation $u+iu_{t}+1/(2m)\Delta u=0$ when the frequency $\xi$ has magnitude less than $m$, but behaves like the wave equation for higher frequencies.Some basic Strichartz estimates here are in MsSrWa1980; see for instance [Na-p], [MacNaOz-p], [MacNkrNaOz-p] for more recent treatments.

For inhomogeneous estimates it is known that a solution with zero initial data and forcing term containing s-1 derivatives in a dual space ${L_{t}^{Q'}}{L_{x}^{R'}}$ will lie in $L_{t}^{q}L_{x}^{r}$ if both $(q,r)$ and $(Q,R)$ are admissible in the above sense, if $s\geq 0$, and if one has the scaling condition

$1/Q+d/R+1/q+d/r=d+s+1$.The \u201cs-1\u201d represents a smoothing effect of one derivative, though this full gain is only attainable if one uses the energy exponents $L_{t}^{1}L_{x}^{2}$ and ${L_{t}^{\infty }}{L_{x}^{2}}$. It is possible to obtain inhomogeneous estimates in which only one of the exponents are admissible; this phenomenon was first observed in Har1990, Ob1989 (see also KeTa1998).More recently in [Fc-p2], inhomogeneous estimates are obtained with the above scaling condition assuming the weaker conditions $1\leq q,r\leq \infty$ and $1/q<(n-1)(1/2-1/r)$ or $(q,r)=(infty,2)$ and similarly for $Q,R$, and if the following additional conditions hold:

In $d=1,2$ no further conditions are required;

When ,math>d=3 r, R</math> are required to be finite;

When $d>3$, either $1/q+1/Q<1$, $(n-3)/r\leq (n-1)/R$, and $(n-3)/R\leq (n-1)/r$, or $1/q+1/Q=1$, $(n-3)/r<(n-1)/R,(n-3)/R<(n-1)/r,r\geq q$, and $R\geq Q$.

Strichartz estimates extend to situations in which there is a potential or when the metric is variable.For local-in-time estimates and smooth potentials or

metrics this is fairly straightforward (the potential can be treated by iterative methods, and the metric by parametrix methods).More interesting issues

arise for global-in-time estimates with smooth potentials/metrics or local-in-time estimates with rough potentials/metrics (the two types of results are linked by

scaling).For potentials of power-type decay, the global results are as follows:

For potentials of the form $V=a/|x|^{2}$ with $d\geq 3$ and $a>-(d-2)^{2}/4$, one has global Strichartz estimates [BuPlStaTv-p]; a simplified proof and more general result dealing with inverse square-like potentials which are not too negative is in [BuPlStaTv-p2]. The condition on a is necessary to avoid bound states.For potentials decaying slower than this, Strichartz estimates can fail (Duyckaerts?) For potentials decaying an epsilon faster than $1/|x|^{2}$ and assumed to be nonnegative, dispersive and Strichartz estimates were obtained when $d=3$ in GeVis2003.

(More results to be added in future).